Università degli Studi di Padova
Dipartimento di Scienze Statistiche
Corso di Laurea Triennale in Statistica Economia e Finanza
RELAZIONE FINALE
“On the impact of economic policy uncertainty shocks
on macroeconomic expectations in the United States"
Relatore: Prof. Efrem Castelnuovo
Dipartimento di Scienze Economiche e Aziendali ‘Marco Fanno’
Laureanda: Lovato Clara Matricola: 1010635 ANNO ACCADEMICO 2012/13
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Alla mia famiglia che c’è sempre…
A papà e i nostri silenzi
A mamma e la sua tenacia
A Camilla c la sua dolcezza
A Edoardo e la nostra complicità
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« L'incertezza è l'habitat naturale della vita umana, sebbene la speranza di sfuggire ad essa sia il motore delle attività umane. Sfuggire all'incertezza è un ingrediente fondamentale, o almeno il tacito presupposto, di qualsiasi immagine composita della felicità. È per questo che una felicità autentica, adeguata e totale sembra rimanere costantemente ad una certa distanza da noi: come un orizzonte che, come tutti gli orizzonti, si allontana ogni volta che cerchiamo di avvicinarsi a esso. »
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INDEX
Introduction…..………..………...………7
Section 1: describes in more detail the VARIABLES we use to construct our VAR model and policy-related uncertainty index. 1.1 S&P500……….……….………...…….9
1.2 EPU……….……….…….……….9
1.3 CPI AND INFLRATE…..……….……….15
1.4 GDP……….…18
1.5 FFR AND INTEREST RATE………18
Section 2: VAR MODEL THEORY 2.1 TRADITIONAL APPROACHES………..…...21
2.1.1 Structural model 2.1.2 Critique of Lucas 2.1.3 LSE approach 2.2 SIMS: VAR MODEL…..……….………...22
2.2.1 Advantages 2.2.2 Disadvantages 2.2.3 Applications 2.3 VAR MODEL STRUCTURE……….………24
2.4 STRUCTURAL ANALYSIS………..27 2.4.1 Impulse response functions
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Section 3: my ANALYSIS reporting mostly focus on estimates for the dynamic responses of aggregate economic outcomes for the baseline model to policy-related uncertainty shocks.
3.1 PRELIMINARY DESCRIPTIVE ANALYSIS……….……….30
3.1.1 Graphics 3.1.2 Descriptive stats 3.1.3 Correlation 3.2 VAR MODEL……….32
3.2.1 Lag order Selection criteria 3.2.2 Stability and stationarity 3.3 RESIDUAL TESTS……….35
3.3.1 Residual graphics 3.3.2 White Heteroskedasticity test 3.3.3 Correlograms 3.3.4 Autocorrelation (test LM) 3.4 IMPULSE RESPONSE FUNCTIONS………..38
3.5 VARIANCE DECOMPOSITION OF THE FORECAST ERROR…..………...40
Section 4: considers several ROBUSTNESS TESTS and comparisons for my VAR model to find accordance and persistence with macroeconomic theory. 4.1 VAR(4) with 4 log-variables.………42
4.2 VAR(4) baseline with exogenous FFR………45
4.3 VAR(4) baseline with endogenous FFR and S&P 500……….50
4.4 VAR(4) baseline with exogenous FFR and S&P 500……….52
Conclusions……….55
Appendix………..…57
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INTRODUCTION
“The economy, you know, is a dismal science. A science in which the forecasts have the power to determine the facts, and this happens with even greater consequences, even in cases where their predictions are impossible to formulate, and the market is groping in the darkness of uncertainty. Uncertainty is exactly what seems to be ailing the world economy today. What we are witnessing is the overlapping of a financial and economic instability, which are interwoven in an unstable geopolitical scenario, making the situation more and more difficult to decipher.” Attilio di Battista,
Junior Consultant - Economic Research at International Trade Centre
“L’economia, si sa, è una scienza triste. Una scienza in cui le previsioni hanno il potere di determinare i fatti; ciò vale, con conseguenze anche maggiori, anche nei casi in cui proprio le previsioni sono impossibili da formulare, ed il mercato brancola nel buio dell’incertezza. Proprio di questo sembra essere malata oggi l’economia mondiale: di incertezza. Ciò a cui oggi assistiamo è il sovrapporsi di un’instabilità economica e di una finanziaria, che si intrecciano in uno scenario geopolitico instabile, rendendo la situazione sempre più difficile da decifrare.”
We officially entered the global economic crisis in the first quarter of 2008; this crisis continues to be a burden on the world economy to this day. The causes, that trigger this world economy’s condition, are to be found largely in the financial crisis that hit the United States in the third quarter of 2007. The challenging problem of subprime, loans made by U.S. banks to risky borrowers who were unable to meet their mortgage repayments. In addition to this there have been a series of price increases, starting from raw materials, primarily oil, followed by the other fossil fuels, up to food including wheat and rice. Global inflation increased considerably too and a credit crisis developed causing a lack of trust in the financial markets.
In 2009 the industrial crisis made the GDP, of many countries (mainly Western), fell dramatically causing their entrance into recession. A rapid succession of negative concatenated events followed this situation, from which still nowadays the global economy is trying to come out with great difficulty. It triggered a search of Scott Baker, Nick Bloom and Steven J. Davis (2013) to understand the difficulties of an economic recovery in the USA. They identified a factor known as the uncertainty of economic policy, which concerns how managing difficult choices to make expenditures, loans and investments especially, for economic subjects, families and businesses, without economic certainties. This uncertainty is due to politicians’ misguided or unsafe choices in terms of health care, taxation, commercial and financial operations, pushing more and more towards a risk-averse mentality. According to their idea, the lack of security in the micro and macro-economic world causes a vicious cycle where fewer resources are invested in innovation production, less business staff recruited that increases unemployment, and in addition more and more families have turned to ‘hiding their money under the mattress'. This prevents from laying the groundwork for an effective growth and it does not concern only the present but also the
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long run. With their study, the three researchers have created an index that is able to measure the uncertainty of U.S. economic policy in order to understand the significant impact on the economic cycle and maybe be able to predict the effects and changes.
I am interested in how aggregate of output, interest rate, inflation and stock-market index respond to movements in policy-related economic uncertainty. Here I adopt a simple empirical approach to this question, using Vector Auto Regressions (VAR) and simple identifying assumption to estimate the effects of policy uncertainty on aggregate outcomes. I fit a VAR and recover orthogonal shock using Cholesky decomposition with the following ordering: the log of the S&P 500 index, the policy uncertainty index, the consumer price index to control the inflation, the real gross domestic product, the federal funds rate to control the interest rates. In my baseline specification, I run the VAR on quarterly-grow-rate data with four quarterly lags. This approach identifies dynamic relationships among the variables using the Cholesky ordering and differences in timing of movements in the variables.
The estimated effects of political uncertainty on output, inflation and interest rate and stock-market are robust to several modifications to the baseline VAR specification: a VAR(4) with 4 growth-rate variables. In the first robustness check I consider a VAR(4) with log variables comparing it with the baseline one. As a second robustness test I try to consider exogenous the FFR variable instead of endogenous focusing the attention on the possible variations and reactions of the growth-rate of GDP. Its controls what it will happen if the Fed decides to not react turning down the interest rate. As a third test I consider two models VAR(4) with an added variable; the stock-market index S&P500. These models differ for the variable FFR, firstly it is considered exogenous and secondly endogenous.
Therefore, I conduct a VAR analysis, using Cholesky orderings to construct orthogonal shocks and the policy-related uncertainty index to investigate its role as one potential driver of the real economic variables such as inflation, interest rate and GDP. I find that a policy uncertainty shock foreshadows drops of 10% in interest rate after 40 quarters (10 years) and GDP reductions of 16% within 40 quarters. These findings reinforce concerns that policy-related economic uncertainty played a role in the slow growth and fitful recovery of recent years, and they invite further research into the effect of policy-related uncertainty on economic performance.
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Section 1: DATA AND VARIABLES
The time series, that make up the vector autoregressive model, have quarterly frequency and cover the following macroeconomic variables USA:
1. S&P500
2. Economics Policy uncertainty index EPU
3. Consumer price index for all urban consumer, all items CPIASUCSL 4. Real gross domestic product GDPC1
5. Effective federal funds rate FFR
They cover this time range: 1985-01-01 to 2008-04-01. From the first quarter of 1985 to the second of 2008 , when the economic crisis was exploding.
1.1 S&P500
Widely regarded as the best single gauge of the U.S. equities market, this world-renowned index includes 500 leading companies in leading industries of the U.S. economy. Although the S&P 500® focuses on the large cap segment of the market, with approximately 75% coverage of U.S. equities, it is also an ideal proxy for the total market. S&P 500 is part of a series of S&P U.S. indices that can be used as building blocks for portfolio construction. The S&P 500 was built by Standard & Poor's in 1957 and follows the trend of a stock basket formed by the 500 U.S. companies with the largest capitalization. The weight given to each company is directly proportional to the market value of the same. This index is the most widely used to measure the performance of the U.S. equity market and is now recognized as a benchmark for the performance of the portfolio. The Future on S&P 500 introduced in 1982, is the main tool used by managers to follow the index or to hedge the U.S. market. It is contracted at the CME (Chicago Mercantile Exchange).
1.2 Economic Policy Uncertainty index EPU
Uncertainty about tax, spending, monetary and regulatory policy slowed the recovery from the 2007-2009 recession. To measure policy-related economic uncertainty and to estimate the dynamic relationship between output, investment and employment this EPU index was built from three types of components. One component quantifies newspaper coverage of policy-related economic uncertainty. A second component reflects the number of federal tax code provisions set to expire in future years. The third component uses disagreement among economic forecasters over a future federal government purchases and the future CPI price level as a proxy for uncertainty.
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Firstly, EPU index is a new measure and a good proxy for actual policy uncertainty and we can have its evolution since 1985. Secondly, as my thesis’s aim, I estimate the dynamic response to policy-related uncertainty shocks on economic activity in simple vector autoregressive (VAR) models. The VAR estimates show that an innovation (shock) in policy uncertaintyis followed by a decline of about 16% in real GDP (from variance decomposition of my baseline VAR model) within 40 quarters. However, the VAR results show that increases in our policy-related economic uncertainty index foreshadow declines in output, investment and employment. Many measures of uncertainty rise in recession and fall in recoveries, suggesting that uncertainty could play an important role in driving business shocks. It spikes near consequential presidential elections and major events such as the Gulf wars and the 9/11 attack. It also rises steeply from 2008 onward, as we can see from Figure 1. Some intuitions behind the depressing effect of uncertainty goes back at least to Bernanke (1983). He points out that an high uncertainty gives firms an incentive to delay investment and employment decisions. If every firm waits to invest or hire, the economy contracts generating a recession. When uncertainty falls back down, firms start hiring and investing again to address pent-up demand.
Recently, many commentators have argued that policy-related uncertainty has been a key factor slowing the recovery from the recession of 2007-2009. The claim is that businesses and households are uncertain about the future taxes, spending levels, regulations, health-care reform, and interest rates. In turn, this uncertainty leads them to postpone spending on investment and goods’ consumption and to slow hiring, impeding the recovery.
Nowadays the world's stock markets do not react to news that comes from the economic world, but they look more at the political sphere. That, unfortunately, is not able to give certainty to the markets neither in the United States nor in Europe. And this not only slows down the recovery today, but also weakens the long-term growth. The most striking feature of the current stock market volatility is that the politicians are making news. Their actions and statements regarding bailouts, budget and reforms of the regulatory framework determine the fluctuations of the markets.
This is not normal. Before the financial crisis of 2008, the economic news influenced the financial markets ’performance. A growth in GDP and positive data about employment blew the markets. Negative corporate results caused the stock market crash. Today, unfortunately, the politicians fail to agree generating a broad economic uncertainty. According to our new index, in the 2012 the political uncertainty was close to its all-time highs (Figure 1). Uncertainty is one of the main factors that slow the recovery and threatens to cause a new recession.
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Figure 1. Economic policy uncertainty index in United States. Source: Baker, Bloom e Davis (2011), “Measuring Economic Policy Uncertainty”, Chicago & Stanford mimeo.
This graph displays the Policy-Related Economic Uncertainty index: EPU. We can find spikes in uncertainty corresponding to several well-known prominent events and a substantially higher level of uncertainty since the onset of the Great Recession in 2007. In particular, we find spikes associated with consequential presidential elections, wars, 9/11 attack, contentious budget battles, and a number of spikes during and after the Great Recession. The average index value is 109 in 2006 (the last year before the current crisis) and 233 in the first eight months of 2011 (all-time high); a difference of 124. Uncertainty is considerably higher in the past 10 years than in the previous 15 years.
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Figure 1. Economic policy uncertainty index in Europe.
From this figure we can compare the USA policy uncertainty index (Figure 1) with the European one (Figure 6); discovering that some peaks are common for both countries, while others are typical of Europe and its major financial and political events.
HOW TO MEASURE POLICY UNCERTAINTY?
Baker, Bloom and Davis have constructed an index of political uncertainty using three types of information: the frequency of newspaper articles about the economic uncertainty and the role of policy, the number of federal provisions in the tax due in the next years and the extent of disagreement between economic forecasts regarding the expected inflation and the purchase of goods and services by the government. Their index shows peaks during the period of uncertainty around major elections, wars and terrorist attacks of September 11. More recently, it peaked after the failure of Lehman Brothers in September 2008, following the approval of the package Tarp. It remained a high value from that moment onwards. Obviously, it is possible that the strong political uncertainty is a consequence of the economic uncertainty. To test this possibility, they use the lists of Google News to build a broad index of economic uncertainty (red line in the Figure 2 below) and a smaller index (blue line), which focuses exclusively on the uncertainty policy. Comparing the two indices (Figure
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2) we can notice the presence of high peaks of economic uncertainty that do not correspond to peaks of political uncertainty. Some examples are: the financial crisis in Asia in 1997 and some periods when it was feared a recession in the second half of the eighties. In summary, the data refute the thesis that economic uncertainty necessarily encourages political uncertainty.
Figure 2. Policy Uncertainty and Economic Policy Uncertainty (overall Economic). Sources: Baker, Bloom e Davis (2011).
WHY THE POLITICY UNCERTAINTY IS SO HIGH?
To identify the reasons for the policy uncertainty, they have deepened the lists of Google News and quantified the mix of factors. Many factors determine the high levels of political uncertainty of 2010-2011, but the monetary and fiscal aspects are the most important. An example is *the tax cuts introduced by George W. Bush about the income, which originally were supposed to expire at the end of 2010. Democrats and Republicans have taken opposing positions on the need to eliminate them or not. Instead of taking a decision in advance of the deadline and eliminate the uncertainty, Congress waited until the last minute to decide to extend the tax breaks. The recent decisions of the Senate on *raising tariffs on imports from China are likely to trigger a trade war. In Europe, the ongoing discussions about *possible bailouts of countries and banks feed the climate of political uncertainty.
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WHY THE POLITICAL UNCERTAINTY IS DANGEROUS?
When companies do not have certainty on taxes, health care costs and the framework of rules assume a cautious position. Making mistakes on investment and hiring are expensive, so many companies expect quieter moments to expand. If too many businesses wait, the recovery does not take off. And low capital investment, product development and training of staff weaken the long-term growth. Baker, Bloom and Davis might expect some improvement in the short term by a stable political system, which was able to increase the certainties? They use simple assumptions and identification vectors of auto regression (for which Sims won the Nobel Prize this year) to estimate the effects of political uncertainty. Their Var for the United States (Figure 12) suggests that bringing political uncertainty to 2006 levels could increase industrial production by 4 percent and create 2.5 million jobs in eighteen months. It is not enough to trigger an economic boom, but it would be a big step forward.
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1.3 Consumer Price Index for All Urban Consumers: CPI and INFLATION RATE The Consumer Price Index (CPI) is a measure of the average change over time in the prices of consumer items goods and services that people buy for day-to-day living. Firstly you have to decide what goods and services included in the average, the CPI follows only the trend of the consumer prices, not taking into account the goods and services not directly purchased by consumers. The CPI is a complex construct that combines economic theory with sampling and other statistical techniques and uses data from several surveys to produce a timely and precise measure of average price change for the consumption sector of the American economy. Production of the CPI requires the skills of many professionals, including economists, statisticians, computer scientists, data collectors, and others. The CPI’s surveys rely on the voluntary cooperation of many people and establishments throughout the country who, without compulsion or compensation, supply data to the Government’s data collection staff.
The Bureau of Labor Statistics (BLS) publishes CPI data every month. The three main CPI series are:
• CPI for All Urban Consumers (CPI-U)
• Chained CPI for All Urban Consumers (C-CPI-U)
• CPI for Urban Wage Earners and Clerical Workers (CPI-W)
The CPI for All Urban Consumers, or CPI-U, which BLS began publishing in January 1978, represents the buying habits of the residents of urban or metropolitan areas in the United States. Each month’s index value displays the average change in the prices of consumer goods and services since a base period, which currently is 1982-84 for most indexes. For example, the CPI-U for March 2002 was 178.8. One interpretation of this is that a representative set of consumer items that cost $100 in 1982-84 would have cost $178.80 in March 2002. The CPI provides an estimate of the price change between any two periods. The percent change between the CPIs for two periods indicates the degree to which prices changed between them. The CPI follows the prices of a sample of items in various categories of consumer spending—such as food, clothing, shelter, and medical services—that people buy for day-to-day living.
The inflation rate, an indicator of the relative change (in time) of the general price level, allows you to see the change in the purchasing power of the currency. It is usually expressed in terms of percentages. Central banks today consider that their main mission is to ensure price stability with the intent to hold the inflation rate low enough, so that there is any abundant concern for anyone.
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The causes of inflation may be different; one of them is determined by the degree to which the increase in the money supply exceeds demand (expansionary monetary policy) that stimulates demand for goods and services and investments. This is a reason that economists have found for price increases in the long run. Other causes can be found in the increase in prices of goods and the increasing cost of imported inputs and intermediate goods. Moreover the increase in cost of inputs also plays a role important to the rising cost of labor.
• INFLATION FROM EXCESS OF CURRENCY
This is the monetarist explanation, which identifies the cause of inflation in the excess of monetary emission with respect to the level required by the volume of transactions. Since the system, according to the monetarists, tends to equilibrium at full employment, any excess money will necessarily release on prices. For monetarists, inflation is due to the errors of the central banks that overly expand the money supply and to excessive government spending.
• DEMAND-PULL INFLATION
This is the Keynesian explanation, which considers the inflation caused by an excess of global demand on global supply. This type of inflation is typical of economies under conditions of full employment. When the inputs are fully employed, an excess of demand over supply causes a general increase in prices, as businesses, searching for workers and raw materials, offer higher wages and prices, spreading in the system the upward pressure on prices. This increase is higher if the difference between aggregate demand and aggregate supply is higher.
• COST-PUSH INFLATION
This explanation, which reflects the conflict between the different social groups in the distribution of income, traces the inflation rise in prices caused by rising production costs, especially those related to labor and raw materials. If costs rise, employers respond by raising prices in order to protect their profits. Of course, the possibility of raising the prices depend on the market regime in which the companies operate. If firms operate under perfect competition, the selling prices cannot be increased, and if they operate in an oligopolistic market, companies can increase selling prices, applying the principle of full cost or mark-up. An explanation of inflation, regarding the category of cost inflation, was proposed in 1958 by the English economist A.W. Philips, who examined the relationship between inflation and unemployment in Britain in the period 1861-1957. The graphical representation of the trade-off between inflation and unemployment is called the Phillips curve. It is a graph that connects the rate of change of money wages (S) and the unemployment rate (D); the unemployed labor force as a percentage of the total.
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The effects of inflation are negative for the whole economic system. There are damages for workers, since, during inflation, the individual prices do not increase uniformly but have a great variability with serious consequences in distribution of income. However, you can limit the damage on workers by automatic indexing mechanisms that allow you to increase wages in relation to the increase in cost of living. The damages are not just for savers but also for the creditors, while the debtors are favored by inflation. The underwriters of government bonds, small investors, holders of insurance or not indexed annuities perceive income that remains nominally unchanged and do not follow the decrease in the purchasing power of the currency. This definitely damages to companies and firms. Entrepreneurs, at least at first, can benefit from the presence of inflationary pressures, it is called annuity by inflation. This advantage does not last because, after a first moment, industrial investments are discouraged since interest rates grow, the difficulty of forecasting and planning inevitably increases, the loss of value of money discourages savings and slows down investments and the formation of new capital. In this area there is also the damage to public finances since, because of inflation, instability tends to spread in the tax system that is not able to obtain immediately the appropriate revenues to public expenditure inflated by the inflation. Inflation then causes damage to the entire system by reducing the export competitiveness. In fact, if prices increase, production costs increase in line with these price increases and this ultimately results in a reduction in exports.
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Inflation is measured in two ways: by means of the Consumer Price Index (CPI), or through the construction of an index of consumer prices. It is one statistical tool that measures changes over time in the prices of a set of goods and services, called the basket, representative of the actual consumption families in a specific year. Another measurement tool is the GDP deflator. The GDP deflator is a tool that allows you to "purify" the growth of GDP by rising prices. Since the Gross Domestic Product is the product, price for quantity, we should know if the growth from one year to another is given by the quantity or produced by rising prices. The deflator is then given by the ratio of Nominal GDP(amount for current prices) and real GDP (constant prices for quantity). Since the value of real GDP is independent of the price dynamics, its changes in value reflect only changes in production economy. Therefore, the GDP is a measure of the production of goods and services. The two indices are moving in the same direction and differ by less than a point percentage.
1.4 Real gross domestic product GDPC1
Gross domestic product (GDP) is the inflation-adjusted measure of the market value of all goods/services produced within the geographical boundaries of the Unites States, regardless of whether the workers/owners are US citizens or not.
GDP is measured as the sum of personal consumption expenditures, gross private domestic investment, net exports of goods and services (exports less imports), and government consumption expenditures and gross investment; GDP = C + I + G + (EX - IMP).
GDP excludes intermediate purchases of goods and services by business.
Real income is the main measure for the material well-being and economic productivity. In my analysis, I use a logarithmic transformation 100 * log(GDP).
1.5 INTEREST RATE and Effective federal funds rate FEDFUNDS/ FFR
The interest rate shows concretely the theoretical price paid by those who receive capital and collected by who offers them. The debtor, receiving a sum of money, agrees to pay a sum greater than the one received. The difference is the interest, which is usually calculated as a percentage of the amount lent. This is the percentage interest rate. The interest rate is variable even in function of the reference currency, the risk related to the solvency of the debtor and the length of the reference period. The data, which I use in my paper, refer to the rate of short-term interest set by the Fed (Federal Reserve, that is the Central Bank of the United States of America), therefore also called the Federal Funds Rate; FFR.
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The Federal funds rate is the interest rate at which banks loan each other overnight funds from their balances with the Federal Reserve.
Expanded Definition: The Federal funds rate is a target rate set by the Federal Reserve for overnight loans between banks. These overnight loans enable banks to maintain enough reserves to meet federal requirements.
The target interest rate does not determine how much it costs to borrow funds overnight; the actual rates are set by the open market. The weighted average of all of these transactions determines the effective rate, which is usually slightly higher than the nominal or target rate. Because of this relationship between the target and the effective rates, changing the Federal funds rate either encourages or discourages banks from raising capital through borrowing. In this way, the Federal Reserve affects how freely the economy operates. The rate of interest on overnight loans of excess reserves made among commercial banks.
Because the Federal Reserve has significant control over the availability of federal funds, the rate is considered an important indicator of Federal Reserve monetary policy and the future direction of other interest rates. A declining federal funds rate may indicate that the Federal Reserve has decided to stimulate the economy by releasing reserves into the banking system. Case Study: The Federal Reserve announced in early December 2001 it was lowering its target federal funds rate from 2.00% to 1.75%, the lowest level in 40 years. The quarter-point decline represented the 11th reduction in the benchmark short-term interest rate since the beginning of the year and established a target rate lower than the rate of inflation. The federal funds rate represents the rate that banks pay to borrow reserves from other banks. This rate influences other short-term rates, including the prime rate and the interest rate on U.S. Treasury bills. The aggressive Federal Reserve policy toward reducing interest rates was intended to stimulate a weak economy that had produced rising unemployment and business failures, especially following the September 11 terrorist attacks in New York City and Washington, D.C.
The Federal Reserve has tools available to affect short-term interest rates but not long-term rates, which are influenced by inflation expectations of lenders and borrowers. Thus, an aggressive policy by the Federal Reserve that reduces interest rates is the main way for the central Bank to stimulate the economy's recovery. Making the dollar more expensive (increasing the rate of interest) causes a reduction in the currency demand by the banking system and thus placing less liquidity in the production system. By doing this, you can keep inflation under control in the growth phase. But now, in times of economic crisis and recession, this would definitely be a suicidal maneuver.
The Federal Reserve Act specifies that the FOMC (Federal Open Market Commitee) should seek "to promote effectively the goals of maximum employment, stable prices, and moderate long-term interest rates." At each meeting, the FOMC closely examines a number of
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indicators of current and prospective economic developments. Then, cognizant that its actions affect economic activity with a lag, it must decide whether to alter the federal funds rate. A decrease in the federal funds interest rate stimulates economic growth, but an excessively high level of economic activity can cause inflation pressures to build to a point that ultimately undermines the sustainability of economic expansion. An increase in the federal funds interest rate will curb economic growth and help contain inflation pressures, and thus can promote the sustainability of an economic expansion, but too large an increase could retard economic growth too much. The Committee's actions on interest rates are undertaken to achieve the maximum rate of economic growth consistent with price stability and moderate long-term interest rates.
The interest rate that banks charge each other for the use of Fed funds. It changes daily and is a sensitive indicator of general interest rate trends. The Fed funds rate is one of the two interest rates set by the Fed, the other being the discount rate. While the Fed can't directly affect this rate, it effectively controls it in the way it buys and sells Treasuries to banks. This is the rate that reaches individual investors, though the changes usually aren't felt for a period of time.
Applying the transformations…
Using the data in its original format could be difficult to interpret. So, after having them turned quarterly, I decided to apply some transformations to make the data more "easily" interpreted, creating new variables most representative or reducing the number of total variables improving the performance of the model
Variable rate: (− ) ⁄ is the variation of an amount compared to the period of the previous survey. If the survey is quarterly, it is the variation of a quarter compared to the previous one. Therefore, we want to highlight the progressive course, the trend and the size. Mathematical transformations: The mathematical functions applied to transform the data are useful to standardize distributions abnormal, trying to linearize a variable. The logarithmic transformations are used to normalize a variable, such as the income, that has an asymmetric distribution. These also tend to reduce the effects of outliers. Taking the logarithm, these variables are turned to normally distribute the data, in this way the result is easy to interpret and sometimes the quality of the results improves too.
1. S&P500 transformation: log(S&P500) 2. EPU
3. CPIASUCSL transformation: (− ) ⁄ INFLRATE
4. GDPC1 transformations:(− ) ⁄ YRATE or 100 ∗ () log(GDP) 5. FFR
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Section 2: VAR MODEL THEORY
In the early 80's, in response to strong criticism addressed to the "structural models" based on systems of simultaneous equations (SES) VAR models were introduced.
2.1 TRADITIONAL APPROACHES
2.1.1 Structural models
Attempt to translate economic relations, based on the theory, deterministic by definition, in statistical equations (i.e. stochastic).
The purpose of these structural models was to estimate empirically the coefficients linking the variables of the economic system, and then answer the following question: ‘what is the effect of an action of the "policy" variables (considered exogenous to the system and under the control of policy makers) on the variables of interest (considered endogenous)?
2.1.2. Critique of Lucas
The economic agents behave in a "forward-looking": that is, the current values of the variables are influenced by expectations about the future of the economy.
These agents adjust their expectations based on the information available.
New economic policies change available information and expectations of the agents and the parameters change accordingly.
Inability to identify the parameters "deep" (deep-parameters) that describe the preferences of consumers and the technology available, the parameters that describe the way in which people form expectations.
2.1.3 LSE approach
Economic theory suggests the general specification of the relevant form of the model, but the precise representation of the PGD (data generating process) is unknown. So, to find the model that best describes the data, the assumed PGD is unknown by definition.
Model in a reduced form is "well specified" in statistical terms. Test empirically assumptions of exogenous variables.
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2.2 SIMS: VECTOR AUTOREGRESSION (VAR) MODEL
With two important articles Sims (1980,1982) introduces VAR models as a response to the "failure" of the traditional one and gave a new approach: starting from a model based on empirical data and on statistical theory, in order to identify the "real" relationships between variables. Some features:
All variables of the economic system are treated as endogenous, there are no prior information derived from economic theory.
The estimated model is "unrestricted", which turns out to be a pure statistical model. From the unrestricted model, some restrictions allow to give an economic
interpretation to the model: structural VAR (SVAR).
VAR models are not intended to describe the whole economy on a large scale, we focus on a limited number of economic variables Y (n × 1 vector).
VAR models are reduced form models: consist of systems of equations that relate the current values of a given set of economic variables with past values of the variables themselves.
All variables assume therefore endogenous nature, while they are only considered exogenous shocks to the system.
The emphasis is more on the statistical properties of the model and its ability to grasp the PGD (data generating process).
There are more sophisticated techniques, which can easily be extended to multivariate analysis, and more structure in our empirical analysis: we can more clearly see the links between empirical and theoretical macroeconomics.
Vector Auto Regressions (VAR) is the dominant research methodology in empirical (time series) macroeconomics. Its goal is the dynamic response of various macro variables to an unexpected exogenous economics policy shock. This is exactly what I want to search in my paper.
2.2.1 Advantages:
The flexibility of the autoregressive formulation allows a statistical description of a wide range of real data sets and provides a unifying framework in which to analyze alternative theories and hypotheses.
2.2.2 Disadvantages:
Such models do not represent the truth in economics but are a useful tool for gaining insight into the interactions between different variables.
Difficult to interpret the estimation results of an unrestricted VAR
Unable to say anything about how the economy reacts to different shocks
Many econometricians consider SVARs as more art than science. One way to assess the robustness of the results is to see whether the impulse responses match our economic intuition and expectations from economic theory.
23 2.2.3 Applications:
The dynamic properties of a VAR(p) are often synthesized through various types of structural analysis. Structural VAR models have four main applications:
1. Impulse response functions (irf): they are used to study the average response of the model variables to a given one-time structural shock.
2. They allow the construction of forecast error variance decompositions that quantify the average contribution of a given structural shock to the variability of the data. 3. They can be used to provide historical decompositions that measure the cumulative
contribution of each structural shock to the evolution of each variable over time. 4. Allow the construction of forecast scenarios conditional on hypothetical sequences
of future structural shocks.
In my analysis I will consider just the first two applications of VAR model: impulse response functions and variance decompositions.
24 2.3 VAR MODEL STRUCTURE
Consider the VAR(1) MODEL:
yt = Φ= Φ= Φ= Φ0000 +Φ +Φ +Φ +Φ1111yt-1 + + + a+ t (1) where
yt= (y1t ,……. ,ykt )T is a stochastic vector (K ×1),
Φ1 is a fixed matrix (K × K) of coefficients,
Φ0 is a vector (K ×1) of intercepts (it allows the possibility of a mean different from zero), at =(a1t,……,akt )T ~ WN(0, ∑) with ∑ not-singular matrix
For example, for K=2 we have
= + , ,+ with at ~ WN(0, ∑) and ∑ = !! ! ! so = + ,+ , + = + ,+ , +
The analysis of the dependences between "#$ and "%$ consists in analyzing the coefficients of the matrix:
Φ1=
and of the covariance matrix:
∑ = !! ! !
In particular the coefficients and measure the dynamic effect between and , while ! the contemporaneous effect.
To see the contemporaneous dependence explicitly, it goes like this:
Apply the triangular decomposition to the positive definite matrix ∑, so
∑=LDLT where L is a lower triangular matrix with the same element (the unit) in the
25 Transform the model in the following way:
L-1yt = L -1 Φ0 + L -1 Φ1yt-1 + L -1 at = Φ0 ∗ + Φ1 ∗ yt-1 + bt
E(bt)=0 and Var(bt)= L-1∑(LT)-1=D with D diagonal, so the components of bt are
uncorrelated.
Given the nature of L-1 (triangularity and unity on the mean diagonal) the k-th
equation of the model becomes:
ykt + ∑()* ())= Φκ0 ∗
+∑ ϕ,)* ()∗ yi,t-1 + bkt
It shows explicitly the contemporaneous relation between ykt and yit , 1≤i≤k-1.
Some recalls:
CHOLESKY DECOMPOSITION Let A be a symmetric and positive definite matrix.
So a unique triangular and lower matrix P exists such that A=PPT.
A = -. / P = 0 √ 0 - √⁄ 2/ − -⁄3 TRIANGULAR DECOMPOSITION
Let A be a symmetric and definitive positive matrix.
So a L triangular lower matrix with unities in the mean diagonal exists such that A=LDLT
and D positive diagonal matrix. A =
-. / L= 1- ⁄ 01 D = 0 / − -0⁄
The triangular decomposition is a particular case of Cholesky decomposition. In fact we can write A=LDLT = L √√ LT = (L √ )(√ L)T=PPT where L √4= P.
L√D = 1 0
- ⁄ 1 0√0 2/ − -0 ⁄3 = P
To apply all the methods of analysis within VAR analysis is required the condition of stationarity of the autoregressive representation.
The VAR(1) model is stable if all the eigenvalues of Φ1 are less than 1 in absolute value.
As the stability condition ensures that the moments up to the second order of the process are independent from t, in this case stability implies also stationarity.
Stationarity condition results |78| <1 i=1,..,K where 9: are solutions of the equation |λIk-Φ1|=0. This equation is equivalent to |Ik-Φ1z|≠0 for each |;| ≤1. Using the lag operator |Φ(z)|≠0 for
each |<| ≤1 as Φ(z)= Ik-Φ1z =0 is the characteristic equation of the model VAR(1). Therefore, if the eigenvalues of Φ1 are less than 1 in absolute value, for j→∞ (MA(∞) form) this equation yt
=Φ0 +Φ1yt-1 + at becomes yt = µ + ∑ ϕ=>* > a@> t=0,±1,±2.... where µ = (Ik-Φ1) -1
26 Consider the VAR(p) MODEL:
yt = Φ= Φ= Φ= Φ0000 + + Φ + +ΦΦΦ1111yt-1 + ...++ ...++ ...+ Φ+ ...+ΦΦΦpyt-p + at (2) where
yt= (y1t ,……. ,ykt )Τ is a stochastic vector (K ×1), Φj j=1,…,p are matrix (K × K) of coefficients, Φ0 is a vector (K x1) of intercepts ,
at =(a1t,……,akt )T~ WN(0, ∑) with ∑non-singular matrix
Using the lag operator Φ(Β)Φ(Β)Φ(Β) yΦ(Β) t = Φ= Φ= Φ= Φ0000 + + a + + t (3)
where the characteristic polynomial is Φ(Β)= Ικ − ΦΦ(Β)= Ικ −Φ(Β)= Ικ −Φ(Β)= Ικ −ΦΦΦ1111ΒΒ −.ΒΒ−...−.−...−...−...−...− ΦΦΦΦpBp (4)
each VAR(p) model can be written in VAR(1) form. So the VAR(p)’s properties can be derived from those of a VAR(1) model. The compact or canonical form is:
yt = A0 +A1 yt-1+bt (5)
where: yt is a (Kp x1)-order-matrix,A0: (Kpx1),A1 : (Kp x Kp),yt-1: (Kp x1) and bt : (Kp x1).
Remembering the results for VAR(1), this VAR(p) model is stable and stationary, if all the eigenvalues of A1 are less than 1 in absolute value, or |Ikp - A1z|≠0 for each |<| ≤1. Moreover
|Ikp-A1z|=|Ικ −Φ1z−....−Φpzp| where Φ(z)= Ικ −Φ1z−....−Φpzp is the characteristic polynomial of VAR(p) model, so the stationarity condition becomes |Ικ −ΦΙκ −ΦΙκ −ΦΙκ −Φ1111z−.−.−...−...−Φ...−Φ.−Φ.−Φpzp |≠≠≠ 0≠ for each |<| ≤1.
The MA(∞) representation of VAR(p) comes from the MA(∞) of VAR(1) as the following: xt= (IKp –A1)-1 A0 + (IKp –A1B)-1b1 =µx + ∑ A=>* >b@>
so: yt = J xt = Jµx+ ∑ J=>* A>JDJb@>= Jµy + ∑ Ψ=>* >a@>
where: Ψi=JA>JD using bt=JDJb@ and Jb@= at..
Introducing the operator ΨΨΨΨ(E)=IK+ΨΨΨΨ1B+ΨΨΨΨ2B2+…=∑ Ψ=8*GΨΨΨFHI
such that ΨΨΨΨ(E)Φ(Φ(Φ(Φ(B)=)=)=)= IK
27 2.4. STRUCTURAL ANALYSIS:
2.4.1 Impulse response functions
Let = the stable VAR, with at~WN(0, ∑) and the not singular matrix ∑ = ∑ ∑
∑ ∑ . So, there is no instantaneous causality between and if and only if ∑12= E(N )=0.
Connecting the uncertainty topic to the VAR model theory, I can create a vector
y@= (EPUt, INFLRATEt, YRATEt, FFRt) and the system:
EPU = α + ∑ β
)* ,) ∆EPU(t − i) + ∑ λ )* ,) INFLRATE(t − i) + ∑ ξ )* ,) ∆YRATE(t − i) + ϕ FFR + a`ab,@
INFLRATE = α + ∑ β
)* ,) ∆EPU(t − i) + ∑ λ )* ,) INFLRATE(t − i) + ∑ ξ )* ,) ∆YRATE(t − i) + ϕ FFR
YRATE = α + ∑ β
)* c,) ∆EPU(t − i) + ∑ λ )* c,) INFLRATE(t − i) + ∑ ξ )* c,) ∆YRATE(t − i) + ϕ FFR
FFR = α + ∑ β
)* ,) ∆EPU(t − i) + ∑ λ )* ,) INFLRATE(t − i) + ∑ ξ )* ,) ∆YRATE(t − i) + ϕ FFR
To study the effect of an uncertainty’s shock, I need to isolate this effect. Suppose that = µ=0 for t<0 and the shock grows by one unit in the period t=0, so a, = a`ab, =1. You want to see what happens to the system in the following periods t=1,2,… in the absence of other shocks, that are a, = ac, = a , =0 and a= a=ac=a =0.
yj=ΦL
Remembering that ΦL=Ψj represents the j-th matrix of coefficients of MA(∞) representation
of VAR(1), that is
yt = ∑ ϕ=J* La@J= ∑ Ψ=J* La@J
then the coefficient of place (i, k) of the matrix ΨLrepresents the expected response of the variable y>,@dJ with respect to a unit change of the variable ye,@. Those coefficients are also called dynamic multipliers.
The result is immediately generalizable to the stationary model VAR(p), recalling that the compact form of a VAR(p) is a VAR(1).
Therefore, be given the VAR(p): yt = Φ0 +Φ1yt-1 + ...+Φpyt-p + at or Φ(Β)yt = Φ0 + at where Φ(Β)= Ικ −Φ1Β−...−ΦpB
p
. Given the stability condition, MA(∞) form is yt =Φ-1(B)Φ0+Φ -1(B) a
@=µ+ ∑ Ψ=>* La@J where the matrix’s coefficients ΨL are obtained recursively from the relation Ψ(B)Φ(B)= IK.
The coefficients Ψ)(,L of the matrix Ψ represent the reaction after j periods of the i-th variable of the system with respect to a unit change in the k-th variable. Those coefficients are the dynamic multipliers.
28 Ψ
Ψ Ψ
Ψgh,I as function of j=0,1,2,… is called impulse response function (irf). Its graphic is very useful to briefly describe the evolution of the response.
If you are interested in the cumulative effect for various periods of a shock in a variable, then you have to consider the matrix sum Sj = ∑ ΨjJ* J. Its elements S)(,k represent the cumulative effect after n periods on the i-th variable in relation to a unit shock of the k-th variable. These quantities are also called intermediate multipliers of order n.
Total cumulative effects for all future periods are obtained from the matrix S==
∑ Ψ= J
J* =Ψ(1)=(Ικ −Φ1−...−Φp)-1. Such effects are also called long-term effects or total multipliers.
Responses to orthogonal impulse
If the components of the error terms a@ are simultaneously related to each other, i.e. ∑ is not diagonal, it is unlikely that the shock which happens to a component remains isolated, but it is easy that a shock in a variable is accompanied by a shock in another variable because of the contemporaneous correlation between components. In this situation it is preferable to orthogonalize the errors and consequently derive the impulse response functions.
For the VAR (1) with K components, zero mean, at ~WN(0, ∑), ∑ singular and
not-diagonal matrix, you have yt = Φ= Φ= Φ= Φ1111yt-1 + at. (1)
Consider the Cholesky decomposition of ∑; ∑ =PPT where P is a lower triangular matrix with positive diagonal elements. Then P-1∑ (P-1)T=I
To get the the irf with orthogonalized errors, take the representation of MA(∞):
y@= a@+ Ψa@+ Ψa@+….. which can write as
y@= PPa@+ ΨPPa@+ ΨPPa@+….. = Θ ε@+ Θε@+ Θε@+…..
where: Θ =P, ΘJ=ΨJP, ε@=P-1a@ and var(ε@)=I.
Pre-multiplying the compact form for B=P-1, you have
B yt = Β= Β= Β= Β1111yt-1 + εεεεt (1* ) where: B1=P -1 Φ1 , εt = P -1 at and εt~ WN(0, I).
The (1*) representation of VAR model, with B≠I and orthogonal errors, is called structural form SVAR, while the (1) representation is reduced form.
29 2.4.2 Forecast error variance decompositions
It allows you to analyze the contribution of innovation of the j-th variable, to the variance of forecast error (h steps ahead) of the k-th variable. Should use the orthogonal errors to identify the contribution.
The forecast error, h step in the future, is:
(y@dm− E[y@dm]) = Θ ε@dm+ Θε@dm+ Θε@dm+…..+Θpε@d The variance of this forecast error h-steps ahead is:
Var(y@dm− E[y@dm])= Θ Ω Θ D + ΘΩ ΘD + ΘΩ ΘD +…+ ΘmΩ ΘmD
The variance decomposition indicates what proportion of the variance of the forecast error for a given variable can be attributed to the different variances Ω. Since the operation makes sense, it is necessary that the total variance of the forecast error is only function of variances and not of covariances. As for the impulse response functions, the variance decomposition requires shocks mutually orthogonal. Since the VAR is a reduced form of a closed system, it is difficult to assume that the residuals of the VAR are mutually orthogonal. Therefore it needs some transformations on the VAR residuals in order to make them orthogonal; considering the structural form we overcome the problem of correlated residuals. The solution proposed by Sims (1980) to the problem of identification is to consider B=I and lower triangular [I-Φ1]-1 , to have exact identification of the VAR.This hypothesis has strong
implications both from the economic point of view and from the statistic point of view. Firstly, we assume that the economy has a recursive structure, secondly we make the impulse response functions and variance decomposition dependent from the arrangement of variables in the VAR. The triangulation (decomposition of Cholesky) is a special case of identification.
30
Section 3: ANALYSIS
In this chapter I analyze the relationships between the variables using the VAR methodology, through which each variable is regressed on p lags of itself and on p lags of the other variables. My data are expressed in the form of time series, so the values vary with respect to a time line, in a sample that goes from the first quarter of 1985 to the second quarter of 2008, the period where there were no real crises yet. The baseline model time series are:
1. EPU
2. CPI transformation: (− ) ⁄ annualized (grow rate) INFLRATE 3. GDPC1 transformation: (− ) ⁄ annualized (grow rate) YRATE 4. FFR
3.1 PRELIMINARY DESCRIPTIVE ANALYSIS 3.1.1 Graphics:
Time series are not seasonal. These series are quite stationary. There is a decreasing trend for FFR. The graph of a time series can be useful at the level intuitive to see if the hypothesis of stationarity applies or not, but it is not a formal test of the assumption of stationarity.
60 80 100 120 140 160 1985 1990 1995 2000 2005 EPU 1 2 3 4 5 6 7 1985 1990 1995 2000 2005 INFLRATE -.002 .000 .002 .004 .006 1985 1990 1995 2000 2005 YRATE 0 2 4 6 8 10 1985 1990 1995 2000 2005 FFR
31 3.1.2 Descriptive statistics
Date: 06/24/13 Time: 15:41
Sample: 1985Q1 2008Q2
EPU INFLRATE YRATE FFR
Mean 96.59667 3.058427 0.003229 4.910222 Median 94.15000 2.954695 0.003242 5.250000 Maximum 144.2000 6.276466 0.005646 9.730000 Minimum 63.40000 1.231950 -0.001079 1.000000 Std. Dev. 20.58962 1.073354 0.001436 2.133087 Skewness 0.528465 0.541177 -0.753677 -0.039488 Kurtosis 2.531131 2.950392 3.258348 2.499871 Jarque-Bera 5.013527 4.402320 8.770717 0.961374 Probability 0.081532 0.110675 0.012458 0.618358 Sum 8693.700 275.2584 0.290573 441.9200 Sum Sq. Dev. 37729.99 102.5358 0.000184 404.9552 Observations 90 90 90 90 3.1.3 Correlation
Covariance Analysis: Ordinary Date: 06/24/13 Time: 15:37 Sample (adjusted): 1986Q1 2008Q2
Included observations: 90 after adjustments Balanced sample (listwise missing value deletion)
Correlation EPU INFLRATE YRATE FFR
EPU 1.000000
INFLRATE 0.245330 1.000000
YRATE -0.338659 -0.244254 1.000000
FFR 0.005365 0.509898 0.187198 1.000000
FFR and INFLRATE are positively and moderately correlated. YRATE-EPU and YRATE-INFLRATE are negatively correlated.
32 3.2 VAR MODEL with grow rate variables
3.2.1 Lag order Selection criteria - Choice of lags
The choice of the lags’ order of the VAR is based on the Akaike information criteria (AIC), the function of which is given by:
q k +
( k
where L is the likelihood, n is the number of observations and k the number of parameters. Since this is a loss function, les is its value and better is the specification choice.
The test results favors a VAR(4).
VAR Lag Order Selection Criteria
Endogenous variables: EPU INFLRATE YRATE FFR Exogenous variables: C
Date: 06/24/13 Time: 16:01 Sample: 1985Q1 2008Q2 Included observations: 86
Lag LogL LR FPE AIC SC HQ
0 -214.8836 NA 0.001909 5.090317 5.204473 5.136260
1 58.63076 515.2248 4.79e-06 -0.898390 -0.327611* -0.668678
2 88.00124 52.59366 3.52e-06 -1.209331 -0.181930 -0.795850*
3 111.5871 40.04109* 2.97e-06* -1.305746 0.098278 -0.788495
4 124.1112 20.09676 3.25e-06 -1.384911* 0.635736 -0.523890
* indicates lag order selected by the criterion
LR: sequential modified LR test statistic (each test at 5% level) FPE: Final prediction error
AIC: Akaike information criterion SC: Schwarz information criterion HQ: Hannan-Quinn information criterion
As we can see from the model’s output and coefficients’ table in the appendix, many coefficients are not significant (p-values in bold are significant) but the
signs are those expected on the basis of economic theory.
Equation: EPU = C(1)*EPU(-1) + C(2)*EPU(-2) + C(3)*EPU(-3) + C(4)*EPU( -4) + C(5)*INFLRATE(-1) + C(6)*INFLRATE(-2) + C(7)*INFLRATE(-3) + C(8)*INFLRATE(-4) + C(9)*YRATE(-1) + C(10)*YRATE(-2) + C(11) *YRATE(-3) + C(12)*YRATE(-4) + C(13)*FFR(-1) + C(14)*FFR(-2) + C(15)*FFR(-3) + C(16)*FFR(-4) + C(17)
The independent variable EPU(-1) and YRATE(-2) and the constant C(17) are significant to explain the dependent variable EPU. Their sign is positive so they accord with the dependent variable EPU sign.
33
Observations: 86
R-squared 0.599972 Mean dependent var 95.52209
Adjusted R-squared 0.507212 S.D. dependent var 20.17046
S.E. of regression 14.15944 Sum squared resid 13833.78
Durbin-Watson stat 2.014444
Equation: INFLRATE = C(18)*EPU(-1) + C(19)*EPU(-2) + C(20)*EPU(-3) + C(21)*EPU(-4) + C(22)*INFLRATE(-1) + C(23)*INFLRATE(-2) + C(24) *INFLRATE(-3) + C(25)*INFLRATE(-4) + C(26)*YRATE(-1) + C(27) *YRATE(-2) + C(28)*YRATE(-3) + C(29)*YRATE(-4) + C(30)*FFR(-1) + C(31)*FFR(-2) + C(32)*FFR(-3) + C(33)*FFR(-4) + C(34)
The independent variable INFLRATE(-1), INFLRATE(-4) and FFR(-1) are significant to explain the dependent variable INFLRATE. The sign of the lagged variable, INFLARATE(-4), is not in accordance with the dependent one, INFLRATE. As we can see later its dynamic response will not be significative. Observations: 86
R-squared 0.812105 Mean dependent var 3.110004
Adjusted R-squared 0.768535 S.D. dependent var 1.060163
S.E. of regression 0.510053 Sum squared resid 17.95060
Durbin-Watson stat 1.763258
Equation: YRATE = C(35)*EPU(-1) + C(36)*EPU(-2) + C(37)*EPU(-3) + C(38)*EPU(-4) + C(39)*INFLRATE(-1) + C(40)*INFLRATE(-2) + C(41) *INFLRATE(-3) + C(42)*INFLRATE(-4) + C(43)*YRATE(-1) + C(44) *YRATE(-2) + C(45)*YRATE(-3) + C(46)*YRATE(-4) + C(47)*FFR(-1) + C(48)*FFR(-2) + C(49)*FFR(-3) + C(50)*FFR(-4) + C(51)
The independent variable YRATE(-1) is significant to explain the dependent variable YRATE and their signs accord to each other.
Observations: 86
R-squared 0.832637 Mean dependent var 0.003199
Adjusted R-squared 0.793828 S.D. dependent var 0.001457
S.E. of regression 0.000662 Sum squared resid 3.02E-05
Durbin-Watson stat 1.987276
Equation: FFR = C(52)*EPU(-1) + C(53)*EPU(-2) + C(54)*EPU(-3) + C(55) *EPU(-4) + C(56)*INFLRATE(-1) + C(57)*INFLRATE(-2) + C(58) *INFLRATE(-3) + C(59)*INFLRATE(-4) + C(60)*YRATE(-1) + C(61) *YRATE(-2) + C(62)*YRATE(-3) + C(63)*YRATE(-4) + C(64)*FFR(-1) + C(65)*FFR(-2) + C(66)*FFR(-3) + C(67)*FFR(-4) + C(68)
The independent variable EPU(-1), EPU(-2), FFR(-1) and FFR(-2) are significant to explain the dependent variable FFR.
Observations: 86
R-squared 0.981471 Mean dependent var 4.821977
Adjusted R-squared 0.977175 S.D. dependent var 2.137020
S.E. of regression 0.322862 Sum squared resid 7.192564
34
3.2.2 Stability/ Stationarity VAR model
To investigate stability and stationarity we have to verify that the roots of the characteristic polynomial are all placed in the unit circle. The model’s estimation responds well to the requirement of stationarity as evidenced by the roots lower than one below and by the unit circle.
Roots of Characteristic Polynomial
Endogenous variables: EPU INFLRATE YRATE FFR Exogenous variables: C Lag specification: 1 4 Date: 06/24/13 Time: 17:27 Root Modulus 0.930191 0.930191 0.845646 - 0.244128i 0.880180 0.845646 + 0.244128i 0.880180 0.725722 - 0.401336i 0.829303 0.725722 + 0.401336i 0.829303 -0.355349 - 0.621726i 0.716112 -0.355349 + 0.621726i 0.716112 0.599442 - 0.118812i 0.611103 0.599442 + 0.118812i 0.611103 -0.526969 0.526969 0.188129 - 0.432507i 0.471651 0.188129 + 0.432507i 0.471651 -0.273179 - 0.321383i 0.421799 -0.273179 + 0.321383i 0.421799 -0.164197 0.164197 0.066485 0.066485
No root lies outside the unit circle. VAR satisfies the stability condition.
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5
35 3.3 RESIDUAL TESTS
It checks the adequacy of the model: the absence of autocorrelation and the presence or absence of heteroskedasticity.
3.3.1 Residual graphics
Variables’ residuals product stationary and White Noise graphs.
3.3.2 White Heteroskedasticity Test
The test regression is run by regressing each cross product of the residuals on the cross products of the regressors and testing the joint significance of the regression. The No Cross Terms option uses only the levels and the squares of the original regresses. The test
regression always includes a constant term as a regressor.
The first part of the output displays the joint significance of the regressors excluding the constant term for each test regression. You may think that each test regression is testing the constancy of each element in the residual covariance matrix separately. Under the null hypothesis of no heteroskedasticity (= homoskedasticity), or no misspecification, the non-constant regressors should not be jointly significant.
-40 -20 0 20 40 88 90 92 94 96 98 00 02 04 06 EPU Residuals -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 88 90 92 94 96 98 00 02 04 06 INFLRATE Residuals -.002 -.001 .000 .001 .002 88 90 92 94 96 98 00 02 04 06 YRATE Residuals -0.8 -0.4 0.0 0.4 0.8 1.2 88 90 92 94 96 98 00 02 04 06 FFR Residuals
36
VAR Residual Heteroskedasticity Tests: No Cross Terms (only levels and squares) Date: 06/28/13 Time: 17:36 Sample: 1985Q1 2008Q2 Included observations: 86 Joint test: Chi-sq Df Prob. 317.6857 320 0.5260
From the joint test, p-value results greater than 0.05. The null hypothesis H0 is accepted;
concluding homoskedasticity.
3.3.3 Correlograms
Displays the pairwise cross-correlograms (sample autocorrelation) for the estimated residuals in the VAR for the specified number of lags (VAR(4)). The graph cross-correlograms displays a matrix of pairwise cross-cross-correlograms. The dotted line in the graphs represents plus or minus two times the asymptotic standard errors of the lagged correlations (computed as 1 √r⁄ ). Here the autocorrelation functions doesn’t exit from the confidence bands for any lags so we can conclude that residuals are distribuited randomly.
-.3 -.2 -.1 .0 .1 .2 .3 2 4 6 8 10 12 Cor(EPU,EPU(-i)) -.3 -.2 -.1 .0 .1 .2 .3 2 4 6 8 10 12 Cor(EPU,INFLRATE(-i)) -.3 -.2 -.1 .0 .1 .2 .3 2 4 6 8 10 12 Cor(EPU,YRATE(-i)) -.3 -.2 -.1 .0 .1 .2 .3 2 4 6 8 10 12 Cor(EPU,FFR(-i)) -.3 -.2 -.1 .0 .1 .2 .3 2 4 6 8 10 12 Cor(INFLRATE,EPU(-i)) -.3 -.2 -.1 .0 .1 .2 .3 2 4 6 8 10 12 Cor(INFLRATE,INFLRATE(-i)) -.3 -.2 -.1 .0 .1 .2 .3 2 4 6 8 10 12 Cor(INFLRATE,YRATE(-i)) -.3 -.2 -.1 .0 .1 .2 .3 2 4 6 8 10 12 Cor(INFLRATE,FFR(-i)) -.3 -.2 -.1 .0 .1 .2 .3 2 4 6 8 10 12 Cor(YRATE,EPU(-i)) -.3 -.2 -.1 .0 .1 .2 .3 2 4 6 8 10 12 Cor(YRATE,INFLRATE(-i)) -.3 -.2 -.1 .0 .1 .2 .3 2 4 6 8 10 12 Cor(YRATE,YRATE(-i)) -.3 -.2 -.1 .0 .1 .2 .3 2 4 6 8 10 12 Cor(YRATE,FFR(-i)) -.3 -.2 -.1 .0 .1 .2 .3 2 4 6 8 10 12 Cor(FFR,EPU(-i)) -.3 -.2 -.1 .0 .1 .2 .3 2 4 6 8 10 12 Cor(FFR,INFLRATE(-i)) -.3 -.2 -.1 .0 .1 .2 .3 2 4 6 8 10 12 Cor(FFR,YRATE(-i)) -.3 -.2 -.1 .0 .1 .2 .3 2 4 6 8 10 12 Cor(FFR,FFR(-i))
37
Following there are the autocorrelation functions of the model residuals. They don’t exit from the confidence bands (±√N) for any delays, thus bringing us to conclude that the residuals are distributed randomly.
3.3.4 Autocorrelation (test LM)
Reports the multivariate LM test statistics for the residual serial correlation up to the specified order. The test statistic for the lag order h is computed by running an auxiliary regression of the residuals ut on the original right-hand regressors and the lagged residual
ut-h , where the missing first h values of ut-h are filled with zero. Under the null hypothesis of
no serial correlation of order h, the LM statistic is asymptotically distribuited X2 with k2
degreed of freedom.
VAR Residual Serial Correlation LM Tests Null Hypothesis: no serial correlation at lag order h
Date: 06/28/13 Time: 17:40 Sample: 1985Q1 2008Q2 Included observations: 86
Lags LM-Stat Prob
1 27.37522 0.0375 2 25.55162 0.0607 3 19.23625 0.2566 4 34.79361 0.0042 5 13.30415 0.6504 6 26.12379 0.0523 7 14.82390 0.5376 8 12.90378 0.6798 9 14.73896 0.5438 10 14.39650 0.5692 11 12.10213 0.7369 12 21.58291 0.1572
Probs from chi-square with 16 df.
Moreover, to test the presence of serial correlation, using the LM test, we can say that the residuals are not autocorrelated. The null hypothesis of absence of correlation is always accepted at any confidence level, except for the 1st delay in which accept to 1% . There is a
problem with the 4th delay that rejects also al'1%. For lag h=1,4 p-values are less than 0.05.
Rejecting the null hypothesis H0, there is serial correlation for lag order h. For the other lags
(h different from 1 and 4) p-values are greater than 0.0h. Null hypothesis H0 is accepted,
